= Variable-permutation action on polynomials
{title2=$(\sigma\cdot f)(x)=f(x_{\sigma(1)},\ldots,x_{\sigma(n)})$}
The <symmetric group> acts on <multivariate polynomials> by replacing each variable $x_i$ by $x_{\sigma(i)}$. With $(\sigma\tau)(i)=\sigma(\tau(i))$, substitution gives $\sigma\cdot(\tau\cdot f)=(\sigma\tau)\cdot f$. Thus it is a left <group action>. A polynomial encoding a combinatorial partition has a <group orbit> encoding all relabelings and a <stabiliser subgroup> encoding its label symmetries.
Back to article page
